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Of the manual flags users can raise, only the "in need of moderator intervention" (custom post flag) can strictly only be handled by a moderator.

However, there are a couple flag types that can only be manually be handled by moderators, but can be actioned by the system if enough users flag. They can only be declined by moderators:

Spam and "rude or abusive" flags (automatic action taken if six people flag, but otherwise can only be handled by a moderator)

Comment flags (comment is automatically deleted if three users flag, more if the comment is upvoted, or instantly after a single flag if the comment contains a "trigger" keyword, but otherwise only moderators can handle this flag)

Additionally, there are flag types that can only be handled by moderators only if the post has already been through a community review. The "not an answer" and "very low quality" flags are sent to community review if the post hasn't been reviewed in the Low Quality Posts queue before (or Triage if it's a Stack Overflow question), or is currently being reviewed there. If the post has already been reviewed by the community, then the flag is sent to moderators and can only be handled by them.

Finally, the several automatic moderator flags raised by the system can also only be handled by moderators, with the exception of the "low quality (auto)" flag, which is handled the same way as a "very low quality" flag from a user, as described above: it goes to community review if the post hasn't already been reviewed by the community.